OCR GCSE Physics A: Deformation, springs and Hooke's law (J249)

This guide covers statements P2.3a to P2.3f of the OCR GCSE (9-1) Gateway Science Physics A (J249) specification, Version 5.0 (July 2026), for first assessment in 2018. They sit in topic P2 Forces, section P2.3 Forces in action. None of these six statements is printed in bold, so they are assessed in both the Foundation Tier and the Higher Tier. Topic P2 is assessed on Paper 1 (Foundation Tier) and Paper 3 (Higher Tier).

Related pages: revision notes, practice questions, the wider OCR GCSE Physics Forces guide, the OCR GCSE Physics hub and the printable checklist. When you want to see where you stand, take a free 10-minute diagnostic.

What this unit covers

Statement	What you must be able to do
P2.3a	Explain that more than one force is needed to stretch, bend or compress an object
P2.3b	Describe the difference between elastic and plastic deformation caused by stretching forces
P2.3c	Describe the relationship between force and extension for a spring and other simple systems, including the graph
P2.3d	Describe the difference between linear and non-linear relationships between force and extension
P2.3e	Calculate a spring constant in linear cases
P2.3f	Calculate the work done in stretching

The specification asks you to recall and apply force exerted by a spring = spring constant × extension, and to apply energy transferred in stretching = ½ × spring constant × extension². In symbols these are F = ke and E = ½ke².

P2.3a: why one force is never enough

A single unbalanced force on a free object changes its motion. It does not change its shape. To deform an object you need forces that pull or push on different parts of it in opposite directions, so the parts move relative to each other.

• Stretching: a spring hangs from a clamp. The weight of the load pulls down on the bottom. The clamp pulls up on the top. Remove the clamp and the whole spring and load simply fall.
• Compressing: squeeze a foam block between your palms. Each hand pushes inwards. If you push the block against a table, the table pushes back up, and the block is squashed between the two.
• Bending: a plank is laid across two bricks. A person’s weight pushes down in the middle. Each brick pushes up near an end. Three forces act, and the plank bows.

Real examples include a climbing rope taking a climber’s weight, a chest expander, a trampoline bed and a diving board.

Worked example 1. Two people each pull one end of a spring with a force of 4.0 N. The spring constant is 250 N/m. Find the extension.

The force that stretches the spring is the tension in it, 4.0 N, not 8.0 N. Each end is pulled with 4.0 N and the pulls are equal and opposite.

e = F / k = 4.0 / 250 = 0.016 m, which is 1.6 cm.

P2.3b: elastic and plastic deformation

Stretching forces cause a distortion of the object.

• Elastic deformation: the object returns to its original shape and size when the forces are removed.
• Plastic deformation: the object does not fully return. It is left with a permanent change of shape.

A spring, a ruler bent slightly and a rubber band stretched gently all behave elastically. Modelling clay, a paper clip opened out far and a spoon bent by a strong person are deformed plastically.

Most materials behave elastically up to a point and plastically beyond it. The loading and unloading comparison in the specification’s practical suggestions (springs and elastic bands loaded with weights, then unloaded) is a direct test. Add weights, then take them off. If the spring or band returns to its natural length, the deformation was elastic. If it stays longer, some of it was plastic.

A spring that has been stretched too far is often permanently longer afterwards. Its force-extension behaviour has also changed, so it is no longer reliable for measuring forces.

P2.3c: force and extension

Extension is the increase in length: extension = stretched length − natural (unstretched) length. A spring that is 24.0 cm long and has a natural length of 20.0 cm has an extension of 4.0 cm.

To investigate a spring, hang it from a clamp beside a vertical ruler. Record the natural length. Add a known weight, wait until it is still, read the new length, and repeat with more weights. Work out each extension, then plot force on the vertical axis against extension on the horizontal axis. This is the kind of investigation the specification suggests (PAG P2).

For a spring the graph has two parts.

1. A straight line through the origin. Force is directly proportional to extension. This is the region where Hooke’s law applies.
2. Beyond a certain point the line bends over. This point is the limit of proportionality. The spring needs less additional force per unit of extension, and if it is taken much further it will not return to its original length.

“Other simple systems” include an elastic band, a rubber cord and a metal wire. A band stretches with a curved graph almost from the start.

Always give the axes with units: force in newtons (N) and extension in metres (m), or centimetres if the question’s graph uses them.

P2.3d: linear and non-linear

• Linear relationship: the graph is a straight line through the origin. Doubling the force doubles the extension.
• Non-linear relationship: the graph is not a straight line. Doubling the force does not double the extension.

A spring is linear up to the limit of proportionality and non-linear beyond it. An elastic band is non-linear throughout most of its range.

Do not confuse the two pairs of words. Linear and non-linear describe the shape of the graph. Elastic and plastic describe whether the object recovers. A rubber band can be non-linear and still elastic. Its unloading curve usually lies below its loading curve, so the two paths are different even though it returns to its starting length.

Worked example 2. Table 1 gives a spring’s behaviour. The natural length is 15.0 cm. Each 50 g mass has a weight of 0.50 N.

Mass (g)	Force (N)	Length (cm)	Extension (cm)
0	0.00	15.0	0.0
50	0.50	17.5	2.5
100	1.00	20.0	5.0
150	1.50	22.5	7.5
200	2.00	25.0	10.0
250	2.50	27.5	12.5
300	3.00	31.0	16.0
350	3.50	36.5	21.5

For the first five loads, every extra 0.50 N gives 2.5 cm more, so the relationship is linear. At 3.00 N the extension jumps by 3.5 cm and at 3.50 N by 5.5 cm, so the line has bent. The limit of proportionality lies between 2.50 N and 3.00 N, close to 2.50 N.

P2.3e: the spring constant

In the linear region:

F = ke, so k = F / e and e = F / k

F is the force in newtons, k is the spring constant in newtons per metre (N/m), and e is the extension in metres. A larger k means a stiffer spring. On a force-extension graph, k is the gradient of the straight section.

Convert extensions to metres before you substitute. 2.5 cm = 0.025 m. 36 mm = 0.036 m.

Worked example 3. A spring of natural length 12.0 cm is 17.0 cm long when a force of 3.0 N pulls on it. Find the spring constant.

1. Extension = 17.0 − 12.0 = 5.0 cm = 0.050 m.
2. k = F / e = 3.0 / 0.050 = 60 N/m.

Worked example 4. Using Table 1, find k from the linear region and the extension at 1.75 N if the spring stayed linear.

1. Take the point (2.00 N, 10.0 cm). The extension is 0.100 m.
2. k = 2.00 / 0.100 = 20 N/m.
3. e = 1.75 / 20 = 0.0875 m, which is 8.75 cm.

Check the answer against the table. 1.75 N lies halfway between 1.50 N and 2.00 N, and 8.75 cm lies halfway between 7.5 cm and 10.0 cm. Good.

Worked example 5. A spring has k = 140 N/m. What force produces an extension of 3.0 cm?

F = ke = 140 × 0.030 = 4.2 N.

The same equation works for compression if the spring is compressed within its linear range. Then e is the compression, the decrease in length.

P2.3f: work done in stretching

When a force stretches a spring, the force does work on it. If the spring behaves elastically, this work is stored as elastic energy in the spring, and the spring returns it when released.

For a spring in its linear region the force rises steadily from 0 to F. The work done equals the area under the force-extension graph, a triangle with base e and height F:

work done = ½ × F × e

Substituting F = ke gives:

E = ½ke²

E is the energy transferred in joules (J), k is in N/m and e is in metres. Square the extension first, then multiply.

Worked example 6. Find the energy stored when the spring in Table 1 is stretched to 2.00 N.

E = ½ × 20 × 0.100² = 0.10 J. As a check, ½ × 2.00 × 0.100 = 0.10 J.

Worked example 7. A spring of k = 48 N/m stores 1.5 J. Find the extension.

1. E = ½ke², so e² = 2E / k = (2 × 1.5) / 48 = 0.0625.
2. e = √0.0625 = 0.25 m.

Doubling the extension quadruples the energy, because e is squared. A spring of k = 300 N/m compressed by 4.0 cm stores ½ × 300 × 0.040² = 0.24 J. Compressed by 8.0 cm it stores 0.96 J.

Non-linear systems

When the graph is a curve there is no simple formula. Work done is still the area between the curve and the extension axis. Find it by counting squares, or by splitting the area into strips and treating each as a trapezium.

Worked example 8. An elastic band is loaded and the results recorded.

Extension (cm)	0	2	4	6	8	10
Force (N)	0	0.8	1.2	1.5	2.1	3.4

Each strip is 2 cm wide. Area of a strip = average force × width.

• 0 to 2 cm: (0 + 0.8) / 2 × 2 = 0.8
• 2 to 4 cm: (0.8 + 1.2) / 2 × 2 = 2.0
• 4 to 6 cm: (1.2 + 1.5) / 2 × 2 = 2.7
• 6 to 8 cm: (1.5 + 2.1) / 2 × 2 = 3.6
• 8 to 10 cm: (2.1 + 3.4) / 2 × 2 = 5.5

Total = 14.6 N cm = 0.146 J, since 100 N cm = 1 J. The work done stretching the band to 10 cm is about 0.15 J. It is an estimate, because the strips are straight-line approximations.

Common errors

• Using the stretched length instead of the extension.
• Leaving the extension in centimetres in F = ke or E = ½ke². The spring constant then comes out 100 times too small, and the energy is wrong by a factor of 10 000.
• Writing ½ × (ke)² instead of ½ × k × e².
• Using E = ½ke² on an elastic band or past the limit of proportionality.
• Saying a single force can stretch a spring with nothing at the other end.
• Calling a curved graph “plastic”. Curved means non-linear.
• Reading k from a point on the curved section instead of the straight section.

Next steps

Condense the unit with the revision notes, then try the practice questions. If you want teacher guidance, book a free trial class.

Official syllabus

OCR, GCSE (9-1) Gateway Science Physics A (J249), Version 5.0 (July 2026), first assessment 2018. Statements P2.3a to P2.3f.